What is the value of sin(600)+sec(660)-cot(690) ?

The value of sin(600)+sec(660)-cot(690) is -(sqrt(3))/2+2+sqrt(3)

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sin(600)+sec(660)-cot(690)

Solution

sin (60 0^{\circ}) + sec (66 0^{\circ}) - cot (69 0^{\circ})

Solution

- \frac{3}{2} + 2 + 3

Decimal Notation

2.86602 \dots

Solution steps

sin (60 0^{\circ}) + sec (66 0^{\circ}) - cot (69 0^{\circ})

sin (60 0^{\circ}) = sin (24 0^{\circ})

sin (60 0^{\circ})

Rewrite

60 0^{\circ}

36 0^{\circ} + 24 0^{\circ}

= sin (36 0^{\circ} + 24 0^{\circ})

Apply the periodicity of

sin

sin (x + 36 0^{\circ}) = sin (x)

sin (36 0^{\circ} + 24 0^{\circ}) = sin (24 0^{\circ})

= sin (24 0^{\circ})

= sin (24 0^{\circ}) + sec (66 0^{\circ}) - cot (69 0^{\circ})

sec (66 0^{\circ}) = sec (30 0^{\circ})

sec (66 0^{\circ})

Rewrite

66 0^{\circ}

36 0^{\circ} + 30 0^{\circ}

= sec (36 0^{\circ} + 30 0^{\circ})

Apply the periodicity of

sec

sec (x + 36 0^{\circ}) = sec (x)

sec (36 0^{\circ} + 30 0^{\circ}) = sec (30 0^{\circ})

= sec (30 0^{\circ})

= sin (24 0^{\circ}) + sec (30 0^{\circ}) - cot (69 0^{\circ})

cot (69 0^{\circ}) = cot (15 0^{\circ})

cot (69 0^{\circ})

Rewrite

69 0^{\circ}

18 0^{\circ} \cdot 3 + 15 0^{\circ}

= cot (18 0^{\circ} 3 + 15 0^{\circ})

Apply the periodicity of

cot

cot (x + 18 0^{\circ} \cdot k) = cot (x)

cot (18 0^{\circ} \cdot 3 + 15 0^{\circ}) = cot (15 0^{\circ})

= cot (15 0^{\circ})

= sin (24 0^{\circ}) + sec (30 0^{\circ}) - cot (15 0^{\circ})

Rewrite using trig identities:

sin (24 0^{\circ}) = 2 sin (12 0^{\circ}) cos (12 0^{\circ})

sin (24 0^{\circ})

Write

sin (24 0^{\circ})

sin (2 \cdot 12 0^{\circ})

= sin (2 \cdot 12 0^{\circ})

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= 2 sin (12 0^{\circ}) cos (12 0^{\circ})

= 2 sin (12 0^{\circ}) cos (12 0^{\circ}) + sec (30 0^{\circ}) - cot (15 0^{\circ})

Use the following trivial identity:

sin (12 0^{\circ}) = \frac{3}{2}

sin (12 0^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

Use the following trivial identity:

cos (12 0^{\circ}) = - \frac{1}{2}

cos (12 0^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= - \frac{1}{2}

Rewrite using trig identities:

sec (30 0^{\circ}) = 2

sec (30 0^{\circ})

Rewrite using trig identities:

\frac{1}{cos ( 30 0 ^{\circ} )}

sec (30 0^{\circ})

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

= \frac{1}{cos ( 30 0 ^{\circ} )}

= \frac{1}{cos ( 30 0 ^{\circ} )}

Rewrite using trig identities:

cos (30 0^{\circ}) = \frac{1}{2}

cos (30 0^{\circ})

Rewrite using trig identities:

cos (18 0^{\circ}) cos (12 0^{\circ}) - sin (18 0^{\circ}) sin (12 0^{\circ})

cos (30 0^{\circ})

Write

cos (30 0^{\circ})

cos (18 0^{\circ} + 12 0^{\circ})

= cos (18 0^{\circ} + 12 0^{\circ})

Use the Angle Sum identity:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= cos (18 0^{\circ}) cos (12 0^{\circ}) - sin (18 0^{\circ}) sin (12 0^{\circ})

= cos (18 0^{\circ}) cos (12 0^{\circ}) - sin (18 0^{\circ}) sin (12 0^{\circ})

Use the following trivial identity:

cos (18 0^{\circ}) = (- 1)

cos (18 0^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= (- 1)

Use the following trivial identity:

cos (12 0^{\circ}) = - \frac{1}{2}

cos (12 0^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= - \frac{1}{2}

Use the following trivial identity:

sin (18 0^{\circ}) = 0

sin (18 0^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 0

Use the following trivial identity:

sin (12 0^{\circ}) = \frac{3}{2}

sin (12 0^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

= (- 1) (- \frac{1}{2}) - 0 \cdot \frac{3}{2}

Simplify

= \frac{1}{2}

= \frac{1}{\frac{1}{2}}

Simplify

= 2

Use the following trivial identity:

cot (15 0^{\circ}) = - 3

cot (15 0^{\circ})

cot (x)

periodicity table with

18 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

= - 3

= 2 \cdot \frac{3}{2} (- \frac{1}{2}) + 2 - (- 3)

Simplify

2 \cdot \frac{3}{2} (- \frac{1}{2}) + 2 - (- 3) : - \frac{3}{2} + 2 + 3

2 \cdot \frac{3}{2} (- \frac{1}{2}) + 2 - (- 3)

Remove parentheses:

(- a) = - a, - (- a) = a

= - 2 \cdot \frac{3}{2} \cdot \frac{1}{2} + 2 + 3

2 \cdot \frac{3}{2} \cdot \frac{1}{2} = \frac{3}{2}

2 \cdot \frac{3}{2} \cdot \frac{1}{2}

Multiply fractions:

a \cdot \frac{b}{c} \cdot \frac{d}{e} = \frac{a \cdot b \cdot d}{c \cdot e}

= \frac{3 \cdot 1 \cdot 2}{2 \cdot 2}

Cancel the common factor:

2

= \frac{3 \cdot 1}{2}

Multiply:

3 \cdot 1 = 3

= \frac{3}{2}

= - \frac{3}{2} + 2 + 3

= - \frac{3}{2} + 2 + 3

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Frequently Asked Questions (FAQ)

What is the value of sin(600)+sec(660)-cot(690) ?
The value of sin(600)+sec(660)-cot(690) is -(sqrt(3))/2+2+sqrt(3)